I don't have children, but I certainly know people who have moved abroad with children.
The best way to do it would be to first find work abroad. Once you have a job offer, you may find that that company pays for your family to move to their country.
You would have to do your research into schooling and so on, but these days, most countries have international schools, where expats send their kids, as they learn the American schooling style, and of course it's in English.
You might find that one of you would need to go to your country of choice, set up interviews and such, and see people in person, in order to get a job.
It does get more difficult as you get older, to find work abroad (as it does in your own country), but it is not impossible.
I know that many companies in the Netherlands employ people from the US, as there is a skills shortage here. My husband is an IT manager at a telecommunications company, and he struggles to find people to fill positions. Half the people at his company are not from here!
If I were you, I would start by browsing all the job sites. You will be surprised at how many jobs ask for people who can speak English. Just apply, apply, apply. You never know!
I hope I have answered some questions for you?